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A132269
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Product{k>=0, 1+floor(n/2^k)}.
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15
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1, 2, 6, 8, 30, 36, 56, 64, 270, 300, 396, 432, 728, 784, 960, 1024, 4590, 4860, 5700, 6000, 8316, 8712, 9936, 10368, 18200, 18928, 21168, 21952, 27840, 28800, 31744, 32768, 151470, 156060, 170100, 174960, 210900, 216600, 234000, 240000, 340956
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OFFSET
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0,2
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COMMENTS
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If n is written in base-2 as n=d(m)d(m-1)d(m-2)...d(2)d(1)d(0) (where d(k) is the digit at position k) then a(n) is also the product (1+d(m)d(m-1)d(m-2)...d(2)d(1)d(0))*(1+d(m)d(m-1)d(m-2)...d(2)d(1))*(1+d(m)d(m-1)d(m-2)...d(2))*...*(1+d(m)d(m-1)d(m-2))*(1+d(m)d(m-1))*(1+d(m)).
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LINKS
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Table of n, a(n) for n=0..40.
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FORMULA
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Recurrence: a(n)=(1+n)*a(floor(n/2)); a(2n)=(1+2n)*a(n); a(n*2^m)=product{1<=k<=m, 1+n*2^k}*a(n).
a(2^m-1)=2^(m(m+1)/2), a(2^m)=2^(m(m+1)/2)*product{0<=k<=m, 1+1/2^k}, m>=1.
a(n)=A132270(2n)=(1+n)*A132270(n).
Asymptotic behavior: a(n)=O(n^((1+log_2(n))/2)); this follows from the inequalities below.
a(n)<=A098844(n)*product{0<=k<=floor(log_2(n)), 1+1/2^k}.
a(n)>=A098844(n)/product{1<=k<=floor(log_2(n)), 1-1/2^k}.
a(n)<c*n^((1+log_2(n))/2)=c*2^A000217(log_2(n)), where c=product{k>=0, 1+1/2^k}=4.7684620580627... (see constant A081845).
a(n)>n^((1+log_2(n))/2)=2^A000217(log_2(n)),
lim sup a(n)/A098844(n)=product{k>=0, 1+1/2^k}=4.7684620580627..., for n-->oo (see constant A081845).
lim inf a(n)/A098844(n)=1/product{k>0, 1-1/2^k}=1/0.288788095086602421..., for n-->oo (see constant A048651).
lim inf a(n)/n^((1+log_2(n))/2)=1, for n-->oo.
lim sup a(n)/n^((1+log_2(n))/2)=product{k>=0, 1+1/2^k}=4.7684620580627..., for n-->oo (see constant A081845).
lim inf a(n+1)/a(n)=product{k>=0, 1+1/2^k}=4.7684620580627... for n-->oo (see constant A081845).
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EXAMPLE
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a(10)=(1+floor(10/2^0))*(1+floor(10/2^1))*(1+floor(10/2^2))*(1+floor(10/2^3))=11*6*3*2=396; a(17)=4860 since
17=10001(base-2) and so a(17)=(1+10001)*(1+1000)*(1+100)*(1+10)*(1+1)(base-2)=18*9*5*3*2=4860.
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CROSSREFS
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Cf. A048651, A081845, A132270, A132271(p=10), A132272, A132327(p=3), A132328.
For formulas regarding a general parameter p (i.e. terms 1+floor(n/p^k)) see A132271.
For the product of terms floor(n/p^k) see A098844, A067080, A132027-A132033, A132263, A132264.
Sequence in context: A056188 A020696 A216762 * A053287 A086323 A075999
Adjacent sequences: A132266 A132267 A132268 * A132270 A132271 A132272
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KEYWORD
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nonn
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AUTHOR
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Hieronymus Fischer, Aug 20 2007
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STATUS
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approved
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